Modified isochronal tests for gas wells

This article discusses the implementation and analysis of the modified isochroncal testing for gas well deliverability tests. Both the Rawlins and Schellhardt and Houpeurt analysis techniques are presented in terms of pseudopressures.

The time to build up to the average reservoir pressure before flowing for a certain period of time still may be impractical, even after short flow periods. Consequently, a modification of the isochronal test was developed to shorten test times further. The objective of the modified isochronal test is to obtain the same data as in an isochronal test without using the sometimes lengthy shut-in periods required to reach the average reservoir pressure in the drainage area of the well.

Modified isochronal test procedure and analysis

The modified isochronal test (Fig. 1) is conducted like an isochronal test, except the shut-in periods are of equal duration. The shut-in periods should equal or exceed the length of the flow periods. Because the well does not build up to average reservoir pressure after each flow period, the shut-in sandface pressures recorded immediately before each flow period rather than the average reservoir pressure are used in the test analysis. As a result, the modified isochronal test is less accurate than the isochronal test. As the duration of the shut-in periods increases, the accuracy of the modified isochronal test also increases. Again, a final stabilized flow point usually is obtained at the end of the test but is not required for analyzing the test data.

• 

  Fig. 1 – Pressure and flow history of a typical modified isochronal test.

The well does not build up to the average reservoir pressure during shut-in; the analysis techniques for the modified isochronal tests are derived intuitively. Recall the transient flow equation, expressed in terms of the reservoir pressure at the start of flow, on which isochronal testing is based:

In new reservoirs with little or no pressure depletion, p s equals the initial reservoir pressure (p_s = p_i); in developed reservoirs, p_s < p_i. In addition, the transient drainage radius, r_d, in Eq. 2 is defined as

....................(1)

....................(2)

Because r_d is a function of time and not of flow rate, Eq. 2 is valid at any fixed time. For modified isochronal tests, use Eq. 2, in which the stabilized shut-in BHP, p_s, is replaced with shut-in BHP, p_ws, measured before each flow period, where p_ws ≤ p_s,

....................(3)

Eq. 3 can be rewritten as

....................(4)

where ....................(5)

and ....................(6)

Eq. 6 indicates that b is independent of time and will remain constant during the test. Similarly, Eq. 5 indicates that a_t is constant for a fixed time. The similarity of Eqs. 2 and 3 for the isochronal and modified isochronal tests, respectively, suggests that the modified isochronal test data can be analyzed like those from an isochronal test.

The theory developed for the modified isochronal test implies that, if the intuitive approximation of using p_ws instead of p_s is valid, the transient data will plot as straight line for each time with the same slope, b. The intercept, a_t, will increase with increasing time. By drawing a line with slope b through the stabilized data point and using the coordinates of the stabilized point and the slope, a stabilized intercept, a, that is independent of time can be calculated, where

....................(7)

To calculate the absolute open flow (AOF) of the well, use the average reservoir pressure, p_s, measured before the test instead of the p_ws value, or

....................(8)

Two variations of the modified isochronal test are considered: tests with a stabilized flow point obtained at the end of the test and tests run without that final point.

Modified isochronal tests with a stabilized flow point

Rawlins-Schellhardt analysis. Recall the empirical Rawlins and Schellhardt equation in terms of transient isochronal test data:

....................(9)

As in the graphical analysis techniques for isochronal tests, plot several trends of data taken at different times during a modified isochronal test. The slope n of each line through points at equal time values will be constant. However, the intercept, log(C_t), is a function of time but not flow rate. Therefore, a different intercept should be calculated for each isochronal test. Use p_p(p_ws) instead of p_p(p_s) in Eq. 9, which gives

....................(10)

The conventional analysis technique for modified isochronal test data is to plot log [p_p(p_ws) − p_p(p_wf )] vs. log (q) for each time, giving a straight line of slope 1/n and an intercept of {−1/n [log(C_t)]}. The Rawlins-Schellhardt analysis procedure for modified isochronal tests with a stabilized flow point is similar to that presented for isochronal tests, except the plotting functions are developed in terms of the shut-in pressure measured immediately before the next flow period. Only the stabilized, extended flow point is plotted in terms of the average reservoir pressure measured before the test, p_s. Example 1 illustrates the procedure.

Houpeurt analysis. As shown previously, the Houpeurt deliverability equation in terms of transient isochronal test data is

....................(11)

For modified isochronal test data, Eq. 11 should be modified with the assumption that p_p(p_ws) can be used instead of p_p(p_s). With this assumption, Eq. 11 becomes

....................(12)

where ....................(13)

and ....................(6)

The form of Eq. 12 suggests that a plot of

will be a straight line with a slope b and intercept a_t. This theory can be extended to the stabilized point, and we can calculate a stabilized intercept, a, using the coordinates of the stabilized point, or

....................(14)

The slope b of the line through the stabilized point should remain the same. In addition, the average reservoir pressure, which is measured before the test, must be used to evaluate the pseudopressure, p_p(p_s) in Eq. 14. Example 1 illustrates the Houpeurt analysis procedure for modified isochronal tests with a stabilized flow point, which is similar to that presented for isochronal tests.

Example 1: Analysis of a modified isochronal test with a stabilized flow point

Using the following data taken from Well 4, calculate the AOF using both Rawlins and Schellhardt and Houpeurt analysis techniques. Assume p_b = 14.65 psia, where p_p(p_b) = 5.093 × 10⁷ psia²/cp. Table 1 gives the test data. h = 6 ft, r_w = 0.1875 ft, ϕ = 0.2714, T = 540°R (80°F), ≈ p_s= 706.6psia, = 0.015cp, = 0.97, = 1.5×10⁻³ psia⁻¹, γ_g = 0.75, S_w = 0.30, c_f = 3 × 10^–6 psia^–1, and A = 640 acres (assume that the well is centered in a square drainage area).

• 

  Table 1

Solution

Rawlins-Schellhardt analysis. Plot

on log-log graph paper (Fig. 2). Table 2 gives the plotting functions. In addition, plot on the same graph the values of Δp_p that corresponds to the stabilized, extended flow point evaluated at p_s.

For each time, construct the best-fit line through the data points. Because the first data points for each isochron do not follow the trend of the higher rate points, they will be ignored for all subsequent calculations.

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  Fig. 2 – Rawlings-Schellhardt plot of modified isochronal test data, Example 1.

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  Table 2

Calculate the deliverability exponent, n, for each line or isochron. For this example, use least-squares regression analysis. For example, at t = 0.5 hours, n₁ = 0.72. Table 3 summarizes the deliverability exponents.

• 

  Table 3

The arithmetic average of the values in Table 3 is

....................(15)

Because 0.5 ≤ ≤ 1.0, determine the stabilized performance coefficient, C, using the coordinates of the stabilized, extended flow point and n = . Note that the pseudopressure used to calculate the stabilized C value is evaluated at p_s measured at the beginning of the test, rather than p_ws. From Eq. 15,

Then,

To determine the AOF graphically draw a line of slope 1/ through the extended flow point, extrapolate the line to the flow rate at Δp_p = p_p(p_s) - p_p(p_b), and read the AOF directly from the graph (Fig. 3).

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  Fig. 3 – Rawlinsh-Schellhardt analysis of modified isochronal test, Example 1.

Houpeurt analysis. Plot

on Cartesian coordinates (Fig 4). In addition, plot the Δpp/q value that corresponds to the stabilized, extended flow point. Table 4 gives the plotting functions. Construct best-fit lines through the modified isochronal data points for each time. The first data point at the lowest rate for each isochron does not fit on the same straight line as the last three rate points and is ignored in subsequent calculations.

• 

  Fig. 4 – Houpeurt plot of modified isochronal test data, Example 1.

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  Table 4

Determine the slopes of the lines, b, for each isochron by least-squares regression analysis of the best-fit lines through the data points. For example, at t = 0.5 hours, b₁ = 9.654 × 10⁵ psia²/cp/(MMscf/D)². Table 5 summarizes the slopes of the isochrons. The average arithmetic values of the slopes in Table 5 is

Calculate the stabilized isochronal deliverability line intercept, a:

Calculate the AOF potential using and the stabilized a value:

Fig. 5 shows the data for this example.

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  Table 5

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  Fig. 5 – Houpeurt analysis of modified isochronal test data, Example 1.

Modified isochronal tests without a stabilized flow point

Because the well is not required to build up to the average reservoir pressure between the flow periods, the modified isochronal approximation shortens test times considerably. However, the test analysis relies on obtaining one stabilized flow point. Under some conditions, environmental or economic concerns prohibit flaring produced gas to the atmosphere during a long production period, thus preventing measurement of a stabilized flow point. These conditions often occur when new wells are tested before being connected to a pipeline.

Two methods have been developed to analyze modified isochronal tests without a stabilized flow point. The Brar and Aziz method was developed for the Houpeurt analysis, while the stabilized C method was developed for the Rawlins and Schellhardt analysis. The stabilized C method requires prior knowledge of permeability and skin factor or determination of these properties using the methods Brar and Aziz proposed for analyzing modified isochronal tests. Both methods require knowledge of the drainage area shape and size.

Brar and Aziz method-Houpeurt analysis

The Brar and Aziz method is based on the transient Houpeurt deliverability Eqs. 16, 17, 6, and p_s, the stabilized BHP measured before the deliverability test.

....................(16)

....................(17)

Rewriting Eq. 17 as

....................(18)

where ....................(19)

and ....................(20)

m′ and c′ can be calculated using regression analysis of Eq. 18. Alternatively, these variables can be computed directly from the slope and the intercept of a plot of a_t vs. log t. Then calculate the permeability from the slope,

....................(21)

Combining Eqs. 19 and 20 yields an equation for the skin factor,

....................(22)

Estimating the AOF potential of the well requires a stabilized value of a. If the drainage area size and shape are known, the gas permeability calculated from Eq. 21 and the skin factor from Eq. 22 can be used to calculate a:

....................(23)

Table 6 gives shape factors for various reservoir shapes and well locations. The stabilized value of a then is used in Eq. 24 to calculate the AOF of the well:

....................(24)

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  Table 6

• 

  Table 6

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  Table 6

Stabilized C method-Rawlins-Schellhardt analysis

Although the Houpeurt equation has a theoretical basis and is rigorously correct, the more familiar but empirically based Rawlins and Schellhardt equation continues to be used and is indeed favored by many in the natural gas industry. The Houpeurt and Rawlins-Schellhardt analysis techniques are combined here to develop a version of the Rawlins-Schellhardt method for analyzing modified isochronal tests. This analysis technique, called the "Stabilized C" method, is derived by equating the stabilized Rawlins and Schellhardt empirical backpressure equation with the stabilized theoretical Houpeurt equation to obtain equations for the deliverability exponent, n, and the stabilized flow coefficient, C, in terms of the Houpeurt flow coefficients, a and b.

To obtain an equation for the exponent n , take the logarithm of both sides of the stabilized Rawlins and Schellhardt empirical backpressure equation ( Eq. 15 ).

....................(25)

n is the slope of a plot of ln(q) vs. ln(Δp_p). Alternatively, note that n can be expressed as the derivative of ln(q) with respect to ln(Δp_p):

....................(26)

....................(27)

Similarly, take the logarithms of both sides of the Houpeurt Eq. 27

....................(28)

and, thus,

....................(28)

....................(29)

and ....................(30)

In Eq. 30, let q be the unique value q_e at which the d ln(Δp_p)/dq values from the Rawlins-Schellhardt and Houpeurt equations are identical. Solving Eq. 30 for this value of q = q_e,

....................(31)

and ....................(32)

Substituting in the Rawlins-Schellhardt equation and noting that, from the Houpeurt equation (Δp_p)_e = aq_e + bq_e²,

....................(33)

Rearranging,

....................(34)

To apply the stabilized C method, it is necessary to assume that the slope, n, of the Rawlins-Schellhardt deliverability plot is constant. This assumption implies that if values of a and b can be calculated for given reservoir properties, a flow rate can be calculated from Eq. 32, at which the change in pseudopressures calculated by the Rawlins-Schellhardt equation is equal to the change in pseudopressure calculated by the Houpeurt equation. The substitution this flow rate into Eq. 34 allows calculation of a stabilized value of C and this value of C can be used to calculate a value of AOF:

....................(35)

The stabilized C method is limited by the need for values of reservoir properties determined separately from the deliverability test analysis. These properties can be estimated either from drawdown or buildup test analysis or from the Brar and Aziz method.

Example 2: Analysis of modified isochronal test without a stabilized data point

The purpose of this example is to compare results obtained from the analysis of a modified isochronal test (see Table 7) with and without an extended, stabilized data point. Calculate the AOF for the following modified isochronal test data without the extended flow point. Use both the Brar and Aziz and the stabilized C methods. Compare these results with the results obtained by using the extended flow point. This example is Well 8. Only the last four flow points from the test are used in the analysis. Reservoir data are summarized here: h = 454 ft, r_w = 0.2615 ft, ϕ = 0.0675, T = 718°R (258°F), p_s ≅ 4,372.6 psia, μ = 0.023 cp, z = 0.87, c_g = 1.69 × 10^–4 psia^–1, γ_g = 0.65, S_w = 0.3, A = 640 acres. C_A = 30.8828 (assume that the well is centered in a square drainage area). In addition, the results from a drawdown test in this well indicate k_g = 4.23 md and s = −5.2.

• 

  Table 7

The Rawlins and Schellhardt analysis with extended flow point gave C = 2.426 × 10^–3, n = 0.54 and q_AOF = 180.1 MMscf/D. The Houpeurt analysis with extended flow point gave a = 1.455 × 10⁶ psia²/cp/MMscf/D, b = 1.774 × 10⁴ psia²/cp/(MMscf/D)², and q_AOF = 205.6 MMscf/D.

Solution

Brar and Aziz method Step 1—Plot

on Cartesian coordinates (Fig. 6). Table 8 gives the plotting functions. Construct best-fit lines through the modified isochronal data points for each time. Although the data are scattered, all flow rates were used for each isochron.

• 

  Fig. 6 – Brar-Aziz analysis of modified isochronal test data, Example 2.

• 

  Table 8

Step 2—Determine the slopes of the lines, b, for each time by least-squares regression analysis. For example, at t = 3.0 hours, b₁ = 1.823 × 10⁴ psia²/cp/(MMscf/D)². Table 8.28 summarizes the slopes for all isochrons. The arithmetic average value of the b values in Table 9 is

Step 3—Using least-squares regression analysis, calculate the transient deliverability line intercepts for each isochronal line. For example, at t = 3.0 hours,

Table 10 gives the intercepts for each isochron.

• 

  Table 9

• 

  Table 10

Step 4—Prepare a graph of a_t vs. log t (Fig. 7) and draw the best-fit line through data. Using all four data points, calculate m′ and c′ of the best-fit line of the plot of a_t vs. log t using least-squares regression analysis. The result is m′ = 3.871 × 10⁵ psia²/(cp-MMscf/D)/cycle and c′ = 3.909 × 10⁵ psia²/(cp-MMscf/D).

• 

  Fig. 7 – Brar-Aziz plot of a_t vs. t.

Step 5—Calculate the formation permeability to gas using the slope of the semilog straight line.

which compares with k_g = 4.23 md estimated from the drawdown test analysis.

Step 6—Calculate the skin factor with Eq. 22.

This value agrees with s = –5.2 estimated from the drawdown test analysis.

Step 7—Calculate the stabilized flow coefficient, a. Assume that the well is centered in a square drainage area with C_A = 30.8828.

Now, calculate the AOF potential using from Step 2 and the stabilized a value calculated in Step 7.

Stabilized Cmethod. Step 1—Plot

vs. q on log-log coordinates (Fig. 8). Table 11 gives the plotting functions. Construct best-fit lines through the data.

• 

  Fig. 8 – Stabilized V analysis of modified isochronal test data, Example 2.

• 

  Table 11

Step 2—Calculate the deliverability exponent, n, for each line. For this example, use the least-squares regression analysis of all points for each isochron. For example, for t = 3.0 hours, n = 0.63. Table 13 summarizes values of the deliverability exponent for each isochron. The arithmetic average slope of the values in Table 12 is

Step 3—Calculate the theoretical value of the Houpeurt coefficient, a, using the permeability and skin factor values calculated previously with the Brar and Aziz analysis (i.e., k_g = 6.6 md, s = –5.0).

Use the average value for the coefficient, b = 1.878 × 10⁴ psia²/(cp-MMscf/D), obtained from the Brar and Aziz analysis.

• 

  Table 12

Step 4—Calculate the rate at which the change in pseudopressure determined with Rawlins-Schellhardt equation equals the change in pseudopressure determined with the Houpeurt equation. Use the average value for the coefficient, b = 1.878 × 10⁴ psia²/(cp-MMscf/D), obtained from the Brar and Aziz analysis, and the a coefficient from Step 3.

Step 5—Calculate the stabilized C value.

Step 6—Calculate the AOF potential of the well using from Step 2.

Table 13 compares the results of the analyses with and without the extended, stabilized flow points. In general, the results are combrble and illustrate the validity of the Brar and Aziz and the stabilized C methods for modified isochronal tests with no extended, stabilized flow point.

• 

  Table 13

Nomenclature

at	=	, transient deliverability coefficient, psia2-cp/MMscf-D
af	=	, depth of investigation along major axis in fractured well, ft
A	=	πafbf , area of investigation in fractured well, ft2
b	=	(gas flow equation)
bf	=	, depth of investigation of along minor axis in fractured well, ft
c	=	compressibility, psi–1
cf	=	formation compressibility, psi–1
cg	=	gas compressibility, psi–1
co	=	oil compressibility, psi–1
ct	=	Soco + Swcw + Sgcg + cf = total compressibility, psi–1
cw	=	water compressibility, psi–1
	=	total compressibility evaluated at average drainage area pressure, psi–1
C	=	performance coefficient in gas-well deliverability equation, or wellbore storage coefficient, bbl/psi
D	=	non-Darcy flow constant, D/Mscf
h	=	net formation thickness, ft
kg	=	permeability to gas, md
Lf	=	fracture half length, ft
m	=	162.2 qBμ/kh = slope of middle-time line, psi/cycle
n	=	inverse slope of the line on a log-log plot of the change in pressure squared or pseudopressure vs. gas flow rate
pp	=	pseudopressure, psia2/cp
ps	=	stabilized shut-in BHP measured just before start of a deliverability test, psia
pw	=	BHP in wellbore, psi
pwf	=	flowing BHP, psi
pws	=	shut-in BHP, psi
q	=	flow rate at surface, STB/D
rd	=	effective drainage radius, ft
re	=	external drainage radius, ft
rw	=	wellbore radius, ft
s	=	skin factor, dimensionless
Sg	=	gas saturation, fraction of pore volume
So	=	oil saturation, fraction of pore volume
Sw	=	water saturation, fraction of pore volume
t	=	elapsed time, hours
T	=	reservoir temperature, °R
Δpp	=	pseudopressure change since start of test, psia2/cp
μ	=	viscosity, cp
	=	gas viscosity evaluated at average pressure, cp
ϕ	=	porosity, dimensionless

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See also

Deliverability testing of gas wells

Isochronal tests for gas wells

Flow-after-flow tests for gas wells

Single-point tests for gas wells

Flow equations for gas and multiphase flow

PEH:Fluid_Flow_Through_Permeable_Media